The curves $C_1:y=1-\cos x,x\in (0,\pi )and{C}_2:y=\frac{\sqrt{3}}{2}\left|x\right|+a$ will touch each other of

Both the curve touch each other of$\sin x=\frac{\sqrt{3}}{2}$

$\Rightarrow x=\frac{\pi }{3},\frac{2\pi }{3}$

Hence point of contact $\left(\frac{\pi }{3},\frac{1}{2}\right)or\left(\frac{2\pi }{3},\frac{3}{2}\right)$

For $\left(\frac{{\displaystyle \pi }}{{\displaystyle 3}},\text{\hspace{0.17em}}\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)$ we get $a=\frac{1}{2}-\frac{\pi }{3\sqrt{3}}$

For $\left(\frac{{\displaystyle 2\pi }}{{\displaystyle 3}},\frac{{\displaystyle 3}}{{\displaystyle 2}}\right)$ we get$a=\frac{3}{2}-\frac{\pi }{\sqrt{3}}$.